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Properties

Label 2-735-105.2-c1-0-5
Degree $2$
Conductor $735$
Sign $-0.504 - 0.863i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.391 − 1.46i)2-s + (0.879 + 1.49i)3-s + (−0.246 + 0.142i)4-s + (−1.82 + 1.29i)5-s + (1.83 − 1.86i)6-s + (−1.83 − 1.83i)8-s + (−1.45 + 2.62i)9-s + (2.60 + 2.15i)10-s + (−0.791 + 0.457i)11-s + (−0.429 − 0.243i)12-s + (−3.07 + 3.07i)13-s + (−3.53 − 1.58i)15-s + (−2.24 + 3.88i)16-s + (−1.16 − 0.311i)17-s + (4.40 + 1.09i)18-s + (−5.95 − 3.43i)19-s + ⋯
L(s)  = 1  + (−0.276 − 1.03i)2-s + (0.507 + 0.861i)3-s + (−0.123 + 0.0712i)4-s + (−0.815 + 0.578i)5-s + (0.749 − 0.762i)6-s + (−0.648 − 0.648i)8-s + (−0.484 + 0.874i)9-s + (0.822 + 0.682i)10-s + (−0.238 + 0.137i)11-s + (−0.124 − 0.0702i)12-s + (−0.854 + 0.854i)13-s + (−0.912 − 0.409i)15-s + (−0.561 + 0.971i)16-s + (−0.281 − 0.0755i)17-s + (1.03 + 0.258i)18-s + (−1.36 − 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207101 + 0.360791i\)
\(L(\frac12)\) \(\approx\) \(0.207101 + 0.360791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.879 - 1.49i)T \)
5 \( 1 + (1.82 - 1.29i)T \)
7 \( 1 \)
good2 \( 1 + (0.391 + 1.46i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.791 - 0.457i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 + (1.16 + 0.311i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.95 + 3.43i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 0.505i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (-2.31 - 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.774 - 0.207i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.922iT - 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (2.71 + 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.85 - 10.6i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.94 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.533 - 0.924i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 - 6.83i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (2.10 + 0.564i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.62 - 1.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 + (-5.64 + 9.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47745156608646260789437724529, −10.16867993488877547039338355011, −9.106623751365704140957954737337, −8.463989010224708491496874034249, −7.26782658841075686444282396830, −6.41995852271778862371846615692, −4.75700808578386484398497417831, −4.02399408154662488794984285212, −2.93436202763576564091370820849, −2.19341213607958414693434650739, 0.19837829718444030588527166754, 2.20624574703408102584740162246, 3.43104745062974525528530721110, 4.83245737418647032197708334221, 5.97665835580513762435259390242, 6.75348358804984574097065657620, 7.72820596973941469797155267898, 8.119670555222180012962778532934, 8.684501377296225278052265483870, 9.785280315018818472523377149940

Graph of the $Z$-function along the critical line